Principal component analysis

Stat 253

Author

Your Name

Context

CONTEXT

In unsupervised learning we don’t have any y outcome variable, we’re just exploring the structure of our data. This can be divided into 2 types of tasks:

  • clustering
    • GOAL: examine structure & similarities among the individual observations (rows) of our dataset
    • METHODS: hierarchical and K-means clustering
  • dimension reduction
    • GOAL: examine & simplify structure among the features (columns) of our dataset
    • METHODS: principal components analysis (and many others, including singular value decomposition (SVD), Uniform Manifold Approximation and Projection (UMAP))


DIMENSION REDUCTION MOTIVATION

Especially when we have a lot of features, dimension reduction helps:

  • identify patterns among the features
  • conserve computational resources
  • feature engineering: create salient features to use in regression & classification (will discuss next class)


PRINCIPAL COMPONENT ANALYSIS (PCA)





INSPIRATION

PCA is pretty cool. Check out this article “Genes mirror geography in Europe” which examined more than 500,000 DNA sites on 3,000 Europeans. Thus we have high dimensional data with 3,000 rows (n) and 500,000 columns (p), thus p > n. We can capture much of the geographic relationship by reducing these 500,000 features to just 2 principal components!





Principal Component Analysis

NOTE

The section here provides a general overview of the PCA algorithm.

The details require linear algebra, which is not a pre-req for this course.

If you’re curious, more details are provided in the Deeper Learning section below.





PRINCIPAL COMPONENT ANALYSIS

Suppose we start with high dimensional data with p correlated features: \(x_1\), \(x_2\), …, \(x_p\). We want to turn these into a smaller set of k < p features or principal components \(PC_1\), \(PC_2\), …., \(PC_k\) that:

  • are uncorrelated (i.e. each contain unique information)
  • preserve the majority of information or variability in the original data

Step 1
Define the p principal components as linear combinations of the original x features. These combinations are specified by loadings or coefficients a:

\[\begin{split} PC_1 & = a_{11} x_1 + a_{12} x_2 + \cdots + a_{1p} x_p \\ PC_2 & = a_{21} x_1 + a_{22} x_2 + \cdots + a_{2p} x_p \\ \vdots & \\ PC_p & = a_{p1} x_1 + a_{p2} x_2 + \cdots + a_{pp} x_p \\ \end{split}\]

The first PC \(PC_1\) is the direction of maximal variability – it retains the greatest variability or information in the original data. The subsequent PCs are defined to have maximal variation among the directions orthogonal to / perpendicular to / uncorrelated with the previously constructed PCs.


Step 2

Keep only the subset of PCs which retain “enough” of the variability / information in the original dataset.





Small Group Discussion

Recall the Australian weather data from Homework 2 (image: http://toursmaps.com/wp-content/uploads/2017/09/australia-map-google-_0.jpg):

# Import the data and load some packages
library(tidyverse)
library(rattle)
data(weatherAUS)

# Note that this has missing values
colSums(is.na(weatherAUS))
##          Date      Location       MinTemp       MaxTemp      Rainfall 
##             0             0          3285          3085          5930 
##   Evaporation      Sunshine   WindGustDir WindGustSpeed    WindDir9am 
##        109530        118202         15766         15659         16135 
##    WindDir3pm  WindSpeed9am  WindSpeed3pm   Humidity9am   Humidity3pm 
##          8668          3842          7254          4351          8329 
##   Pressure9am   Pressure3pm      Cloud9am      Cloud3pm       Temp9am 
##         23000         22981         89076         95092          3310 
##       Temp3pm     RainToday       RISK_MM  RainTomorrow 
##          7310          5930          5929          5929

PCA cannot handle missing values. We could simply eliminate days with any missing values, but this would kick out a lot of useful info. Instead, we’ll use KNN to impute the missing values using the VIM package.

# If your VIM package works, use this chunk to process the data
library(VIM)

# It would be better to impute before filtering & selecting
# BUT it's very computationally expensive in this case
weather_temp <- weatherAUS %>% 
  filter(Date == "2008-12-01") %>% 
  dplyr::select(-Date, -RainTomorrow, -Temp3pm, -WindGustDir, -WindDir9am, -WindDir3pm) %>% 
  VIM::kNN(imp_var = FALSE)

# Now convert Location to the row name (not a feature)
weather_temp <- weather_temp %>% 
  column_to_rownames("Location") 

# Create a new data frame that processes logical and factor features into dummy variables
weather_data <- data.frame(model.matrix(~ . - 1, data = weather_temp))
rownames(weather_data) <- rownames(weather_temp)

If your VIM package doesn’t work, import the processed data from here:

weather_data <- read.csv("https://kegrinde.github.io/stat253_coursenotes/data/weatherAUS_processed_subset.csv") %>% 
  column_to_rownames("Location")



EXAMPLE 1: Research goals

Check out the weather_data:

head(weather_data)
##            MinTemp MaxTemp Rainfall Evaporation Sunshine WindGustSpeed
## Albury        13.4    22.9      0.6         7.4     10.1            44
## Newcastle     13.2    27.2      0.0         7.4     13.0            44
## Penrith       15.2    32.6      0.0         7.4     10.9            59
## Sydney        17.6    31.3      0.0         7.6     10.9            44
## Wollongong     9.5    17.9      0.4         6.8     10.1            52
## Canberra      13.6    25.2      0.0         9.6     13.0            80
##            WindSpeed9am WindSpeed3pm Humidity9am Humidity3pm Pressure9am
## Albury               20           24          71          22      1007.7
## Newcastle             6           19          50          24      1013.9
## Penrith              13           22          35          23      1009.1
## Sydney                2           24          29          21      1009.1
## Wollongong           20           24          52          44      1007.9
## Canberra             26           43          31          28      1006.3
##            Pressure3pm Cloud9am Cloud3pm Temp9am RainTodayNo RainTodayYes
## Albury          1007.1        8        5    16.9           1            0
## Newcastle       1010.1        3        4    21.8           1            0
## Penrith         1007.1        3        4    24.4           1            0
## Sydney          1004.6        3        7    24.9           1            0
## Wollongong      1003.3        5        5    14.0           0            1
## Canberra        1004.4        1        6    19.9           1            0
##            RISK_MM
## Albury           0
## Newcastle        0
## Penrith          0
## Sydney           0
## Wollongong       0
## Canberra         0

  1. Identify a research goal that could be addressed using one of our clustering algorithms.

  2. Identify a research goal that could be addressed using our PCA dimension reduction algorithm.





EXAMPLE 2: Starting small

Let’s start with just 3 correlated features: \(x_1\) (Temp9am), \(x_2\) (MinTemp), and \(x_3\) (WindSpeed9am)

small_example <- weather_data %>% 
  select(Temp9am, MinTemp, WindSpeed9am)

pairs(small_example)

The goal of PCA will be to combine these correlated features into a smaller set of uncorrelated principal components (PCs) without losing a significant amount of information.

  1. The first PC will be defined to retain the greatest variability, hence information in the original features. What do you expect the first PC to be like?

  2. How many PCs do you think we’ll need to keep without losing too much of the original information?





EXAMPLE 3: Defining the PCs

Perform a PCA on the small_example data:

# This code is nice and short!
# scale = TRUE, center = TRUE first standardizes the features
pca_small <- prcomp(small_example, scale = TRUE, center = TRUE)

This creates 3 PCs which are each different combinations of the (standardized) original features:

# Original (standardized) features
scale(small_example) %>% 
  head()
##                Temp9am     MinTemp WindSpeed9am
## Albury     -0.49164877 -0.10016777    0.5179183
## Newcastle   0.25610853 -0.13455372   -1.3350783
## Penrith     0.65287772  0.20930578   -0.4085800
## Sydney      0.72917948  0.62193718   -1.8645059
## Wollongong -0.93419901 -0.77069379    0.5179183
## Canberra   -0.03383817 -0.06578182    1.3120597

# PCs
pca_small %>% 
  pluck("x") %>% 
  head()
##                   PC1         PC2         PC3
## Albury     -0.6119817  0.27733759 -0.26182765
## Newcastle   0.6944764 -1.15487603  0.22381760
## Penrith     0.7312569 -0.09374376  0.30573078
## Sydney      1.7089610 -1.21413476  0.01482615
## Wollongong -1.3091122 -0.09282229 -0.11200583
## Canberra   -0.6683498  1.12907017  0.07404101

Specifically, these PCs are linear combinations of the (standardized) original x features, defined by loadings a:

\(PC_1 = a_{11}x_1 + a_{12}x_2 + a_{13}x_3\)

\(PC_2 = a_{21}x_1 + a_{22}x_2 + a_{23}x_3\)

\(PC_3 = a_{31}x_1 + a_{32}x_2 + a_{33}x_3\)

And these linear combinations are defined so that the PCs are uncorrelated, thus each contain unique weather information about the cities!

pca_small %>% 
  pluck("x") %>% 
  pairs()

  1. Use the loadings below to specify the formula for the first PC:

    PC1 = ___*Temp9am + ___*MinTemp + ___*WindSpeed9am

pca_small %>% 
  pluck("rotation")
##                     PC1       PC2         PC3
## Temp9am       0.6312659 0.2967160  0.71656333
## MinTemp       0.6230387 0.3562101 -0.69637431
## WindSpeed9am -0.4618725 0.8860440  0.03999775
  1. For just the first city, confirm that its PC1 coordinate or score can be calculated from its original coordinates using the formula in part a:
# Original (standardized) coordinates
scale(small_example) %>% 
  head(1)
##           Temp9am    MinTemp WindSpeed9am
## Albury -0.4916488 -0.1001678    0.5179183
    
# PC coordinates
pca_small %>% 
  pluck("x") %>% 
  head(1)
##               PC1       PC2        PC3
## Albury -0.6119817 0.2773376 -0.2618276





EXAMPLE 4: Examining the components of each PC (part 1)

Plots can help us interpret the above numerical loadings, hence the important components of each PC.

# Plot the loadings for all 3 PCs
library(reshape2)
melt(pca_small$rotation[, 1:3]) %>% 
  ggplot(aes(x = Var1, y = value, fill = Var1)) +
    geom_bar(stat = "identity") +
    facet_wrap(~ Var2) + 
    labs(y = "loadings", x = "original features", fill = "original features")

# Focus on the 1st PC (this will be helpful when we have more PCs!)
melt(pca_small$rotation) %>%
  filter(Var2 == "PC1") %>% 
  ggplot(aes(x = Var1, y = value, fill = Var1)) +
    geom_bar(stat = "identity") +
    labs(y = "loadings", x = "original features", fill = "original features")

  1. Which features contribute the most, either positively or negatively, to the first PC?

  2. What about the second PC?





EXAMPLE 5: Examining the components of each PC (part 2)

When we have a lot of features x, the above plots get messy. A loadings plot is another way to visualize PC1 and PC2 (the most important PCs):

  • each arrow represents a feature x
  • the arrow point falls at the feature’s PC1 loading (x-axis) and PC2 loading (y-axis)
  • arrow length reflects how much the feature contributes to the first 2 PCs

It is powerful in that it can provide a 2-dimensional visualization of high dimensional data (just 3 dimensions in our small example here)!

library(factoextra)
fviz_pca_var(pca_small, repel = TRUE)

  1. Positively correlated features point in similar directions. The opposite is true for negatively correlated features. What do you learn here?

  2. Which features contribute the most to the first PC (x-axis)? (Is this consistent with what we observed in the earlier plots?)

  3. What about the second PC?





EXAMPLE 6: Examining the amount of information captured by each PC (numerical metrics)

Now that we better understand the structures of the PCs, let’s examine the relative amount of information they each capture from the original set of features:

# Load package for tidy table
library(tidymodels)

# Measure information captured by each PC
# Calculate variance from standard deviation
pca_small %>% 
  tidy(matrix = "eigenvalues") %>% 
  mutate(var = std.dev^2)
## # A tibble: 3 × 5
##      PC std.dev percent cumulative    var
##   <dbl>   <dbl>   <dbl>      <dbl>  <dbl>
## 1     1   1.50   0.753       0.753 2.26  
## 2     2   0.812  0.220       0.973 0.660 
## 3     3   0.284  0.0268      1     0.0805

NOTE:

  • var = amount of variability, hence information, in the original features captured by each PC
  • percent = % of original information captured by each PC
  • cumulative = cumulative % of original information captured by the PCs
  1. What % of the original information is captured by PC1? Confirm using both the var and percent columns.

  1. What % of the original information is captured by PC2?

  2. In total, 100% of the original information is captured by PC1, PC2, and PC3. What % of the original info would we retain if we only kept PC1 and PC2, i.e. if we reduced the PC dimensions by 1? Confirm using both the percent and cumulative columns.





EXAMPLE 7: Examining the amount of information captured by each PC (SCREE plots)

Especially when we start with lots of features, graphical summaries of the above tidy summary can help understand the variation captured by the PCs:

# SCREE PLOT: % of variance explained by each PC
pca_small %>% 
  tidy(matrix = "eigenvalues") %>% 
  ggplot(aes(y = percent, x = PC)) + 
    geom_point(size = 2) + 
    geom_line() + 
    labs(y = "% of variance explained")

# Cumulative % of variance explained
pca_small %>% 
  tidy(matrix = "eigenvalues") %>% 
  rbind(0) %>% 
  ggplot(aes(y = cumulative, x = PC)) + 
    geom_point(size = 2) + 
    geom_line() + 
    labs(y = "CUMULATIVE % of variance explained")

Based on these summaries, how many and which of the 3 PCs does it make sense to keep? Thus by how much can we reduce the dimensions of our dataset?





EXAMPLE 8: Examining the new PC coordinates of the data points (score plots)

Finally, now that we better understand the “meaning” of our 3 new PCs, let’s explore their outcomes for each city (row) in the dataset. The below scores provide the new coordinates with respect to the 3 PCs:

pca_small %>% 
  pluck("x") %>% 
  head()
##                   PC1         PC2         PC3
## Albury     -0.6119817  0.27733759 -0.26182765
## Newcastle   0.6944764 -1.15487603  0.22381760
## Penrith     0.7312569 -0.09374376  0.30573078
## Sydney      1.7089610 -1.21413476  0.01482615
## Wollongong -1.3091122 -0.09282229 -0.11200583
## Canberra   -0.6683498  1.12907017  0.07404101

A score plot maps out the scores of the first, and most important, 2 PCs for each city. PC1 is on the x-axis and PC2 on the y-axis. Again, since these PCs are linear combinations of all original features (only 3 here), a score plot can provide a 2-dimensional visualization of high dimensional data!

Question: Unless you’re familiar with Australian geography, it might be tough to ascertain any meaningful patterns here. Looking back to the map, and recalling the key information captured by PC1 and PC2, does there appear to be any geographical explanation of which cities are similar with respect to their PC1 and PC2 scores?

# Score plot: plot PC1 scores (x-axis) vs PC2 scores (y-axis) of all data points
fviz_pca_ind(pca_small, repel = TRUE)





EXAMPLE 9: PCA using all features

Repeat the PCA using all 18 original features in the weather_data, our goal being to reduce the dimensions of this dataset while still maintaining a “sufficient” amount of the original information!

pca_weather <- prcomp(weather_data, scale = TRUE, center = TRUE)

This produces 18 uncorrelated PCs that are linear combinations of the original (standardized) features:

pca_weather %>% 
  pluck("x") %>% 
  head()
##                   PC1        PC2         PC3         PC4         PC5        PC6
## Albury     -0.4347320  0.6310084 -0.00249586  1.58334640 -0.75416235  0.5584570
## Newcastle   1.1354168 -1.4874342 -1.68801307 -0.08843211 -0.29711827 -0.3570227
## Penrith     0.5295179 -2.0629031  0.31172583 -0.40979567 -0.55894823  0.2923958
## Sydney      1.0471702 -1.8638332  0.90468479 -0.19738978 -1.36015610 -0.7805383
## Wollongong -2.3244856  1.5540074  0.07924589  0.03387889  0.08171723 -0.2007186
## Canberra   -2.6420711 -2.6213046  0.97957591 -0.87130315 -1.63498832  0.3104203
##                   PC7        PC8         PC9       PC10         PC11
## Albury     -0.5030481 -1.0032728  1.49263281 -0.3770739  0.217419531
## Newcastle  -0.5342649 -0.8260785 -0.12901337  0.5524187 -0.019147225
## Penrith     0.1669854 -0.5264088 -0.03359561  0.0720234  0.003082526
## Sydney     -1.4130771 -1.1624668 -1.05279963 -0.1089470 -0.059738227
## Wollongong  0.5253852 -0.7279453 -0.64889066 -0.3814026  0.264308869
## Canberra   -0.7598967  1.5625904  0.32417904  1.1590349  0.106050706
##                   PC12       PC13          PC14        PC15        PC16
## Albury     -0.91098812  0.3093372 -0.1194868349 -0.04732443  0.20006422
## Newcastle  -0.31768269 -0.3689492 -0.0020756144 -0.25958440 -0.12764212
## Penrith    -0.01413404  0.5423686  0.0751576394  0.11900530 -0.18078633
## Sydney      0.37396249  0.1287991 -0.0270416392  0.30938838  0.06210685
## Wollongong  0.95247931 -0.1039939  0.2175806676 -0.29571978 -0.03130589
## Canberra    0.02913232 -0.3468298 -0.0001989966 -0.05693145  0.10311335
##                   PC17          PC18
## Albury     -0.10798959 -1.356546e-17
## Newcastle   0.14708667  9.745684e-17
## Penrith     0.22254521 -1.245878e-16
## Sydney     -0.09870755 -2.356101e-16
## Wollongong -0.18911254  5.790386e-17
## Canberra   -0.03666081 -2.911212e-16
  1. Suppose we wanted to retain at least 80% of the information in the original 18 features. What’s the smallest number of PCs we need to keep? Thus how much can we reduce the original dimensions?
# Cumulative % of variance explained (in numbers)
pca_weather %>% 
  tidy(matrix = "eigenvalues")
## # A tibble: 18 × 4
##       PC  std.dev percent cumulative
##    <dbl>    <dbl>   <dbl>      <dbl>
##  1     1 2.22e+ 0 0.274        0.274
##  2     2 1.95e+ 0 0.212        0.486
##  3     3 1.62e+ 0 0.146        0.632
##  4     4 1.40e+ 0 0.109        0.741
##  5     5 1.12e+ 0 0.0696       0.810
##  6     6 9.25e- 1 0.0475       0.858
##  7     7 8.32e- 1 0.0385       0.896
##  8     8 6.90e- 1 0.0265       0.923
##  9     9 6.46e- 1 0.0232       0.946
## 10    10 5.68e- 1 0.0179       0.964
## 11    11 4.86e- 1 0.0131       0.977
## 12    12 4.60e- 1 0.0118       0.989
## 13    13 3.05e- 1 0.00517      0.994
## 14    14 2.33e- 1 0.00301      0.997
## 15    15 1.71e- 1 0.00163      0.998
## 16    16 1.28e- 1 0.00091      0.999
## 17    17 1.04e- 1 0.0006       1    
## 18    18 1.34e-16 0            1
# Cumulative % of variance explained (plot)
pca_weather %>% 
  tidy(matrix = "eigenvalues") %>% 
  rbind(0) %>% 
  ggplot(aes(y = cumulative, x = PC)) + 
  geom_point(size = 2) +
  geom_line() + 
  labs(y = "CUMULATIVE % of variance explained")

  1. Check out some plots of the loadings in the more important PCs. Describe the general properties of the first 2 PCs. What original features do they most capture?
# Plot the loadings for first 5 PCs
# We have to use a different color palette -- we need enough colors for our 18 features
pca_weather$rotation %>% as.data.frame() %>% select(PC1:PC5) %>% rownames_to_column(var = 'Variable') %>% pivot_longer(PC1:PC5 ,names_to = 'PC', values_to = 'Value') %>% #melt
  ggplot(aes(x = Variable, y = Value, fill = Variable)) +
    geom_bar(stat = "identity") +
    facet_wrap(~ PC) + 
    labs(y = "loadings", x = "original features", fill = "original features") + 
    scale_fill_manual(values = rainbow(18)) + 
    theme(axis.title.x=element_blank(),
        axis.text.x=element_blank(),
        axis.ticks.x=element_blank())

# Loadings plot: first 2 PCs
fviz_pca_var(pca_weather, repel = TRUE)

  1. Finally, check out the score plot for the first 2 PCs. Does there appear to be any geographical explanation of which cities are similar with respect to their PC1 and PC2 scores?
fviz_pca_ind(pca_weather, repel = TRUE)


EXAMPLE 10: Drawbacks

  1. In what kind of scenario would you be relieved to have PCA as a tool?
  2. What are the drawbacks of PCA? When might you hesitate to use it?


Exercises

Do exercises 7 and 8 in HW6.

The R code section below will be handy.


Notes: R code

Suppose we have a set of sample_data with multiple feature columns x, and (possibly) a column named id which labels each data point.

# Install packages
library(tidyverse)


PROCESS THE DATA

If there’s a column that’s an identifying variable or label, not a feature of the data points, convert it to a row name.

sample_data <- sample_data %>% 
  column_to_rownames("id")

PCA can’t handle NA values! There are a couple options.

# Omit missing cases (this can be bad if there are a lot of missing points!)
sample_data <- na.omit(sample_data)

# Impute the missing cases
library(VIM)
sample_data <- sample_data %>%
  VIM::kNN(imp_var = FALSE)

IF you have at least 1 categorical / factor feature, you’ll need to pre-process the data even further.

You should NOT do this if you have quantitative and/or logical features.

# Turn categorical features into dummy variables
sample_data <- data.frame(model.matrix(~ . - 1, sample_data))

RUN THE PCA

# scale = TRUE, center = TRUE first standardizes the features
pca_results <- prcomp(sample_data, scale = TRUE, center = TRUE)

CHECK OUT THE PCs

# Get the loadings which define the PCs
pca_results %>% 
  pluck("rotation")

# Plot loadings for first "k" PCs (you pick k)
library(reshape2)
pca_results$rotation %>% as.data.frame() %>% 
  select(PC1:PCk) %>% 
  rownames_to_column(var = 'Variable') %>% 
  pivot_longer(PC1:PC5 ,names_to = 'PC', values_to = 'Value') %>% 
  ggplot(aes(x = Variable, y = Value, fill = Variable)) +
    geom_bar(stat = "identity") +
    facet_wrap(~ PC) + 
    labs(y = "loadings", x = "original features", fill = "original features") + 
    theme(axis.title.x=element_blank(),
        axis.text.x=element_blank(),
        axis.ticks.x=element_blank())

# Plot loadings for just the first PC
pca_results$rotation %>% as.data.frame() %>% 
  select(PC1:PCk) %>% 
  rownames_to_column(var = 'Variable') %>% 
  pivot_longer(PC1:PC5 ,names_to = 'PC', values_to = 'Value') %>%
  filter(PC == "PC1") %>% 
  ggplot(aes(x = Variable, y = Value, fill = Variable)) +
    geom_bar(stat = "identity") +
    labs(y = "loadings", x = "original features", fill = "original features")

# Loadings plot for first 2 PCs
library(factoextra)
fviz_pca_var(pca_results, repel = TRUE)



EXAMINE AMOUNT OF VARIABILITY / INFORMATION CAPTURED BY EACH PC

# Load package for tidy table
library(tidymodels)

# Numerical summaries: Measure information captured by each PC
pca_results %>% 
  tidy(matrix = "eigenvalues")

# Graphical summary 1: SCREE PLOT
# Plot % of variance explained by each PC
pca_results %>% 
  tidy(matrix = "eigenvalues") %>% 
  ggplot(aes(y = percent, x = PC)) + 
    geom_point(size = 2) + 
    geom_line() + 
    labs(y = "% of variance explained")

# Graphical summary 2: Plot cumulative % of variance explained by each PC
pca_results %>% 
  tidy(matrix = "eigenvalues") %>% 
  rbind(0) %>% 
  ggplot(aes(y = cumulative, x = PC)) + 
    geom_point(size = 2) + 
    geom_line() + 
    labs(y = "CUMULATIVE % of variance explained")



EXAMINE THE SCORES, i.e PC COORDINATES FOR THE DATA POINTS

# Numerical summary: check out the scores
pca_results %>% 
  pluck("x")

# Graphical summary: Score plot
# Plot PC1 scores (x-axis) vs PC2 scores (y-axis) of all data points
fviz_pca_ind(pca_results, repel = TRUE)







Deeper learning (OPTIONAL)

ALTERNATIVES TO PCA

For more dimension reduction techniques, check out:

  • https://juliasilge.com/blog/billboard-100/
  • https://www.analyticsvidhya.com/blog/2018/08/dimensionality-reduction-techniques-python/



MATH BEHIND PCA

Let \(X\) be our original (centered) \(n \times p\) data matrix. Mathematically, PCA produces an orthogonal linear transformation of \(X\). To this end, first note that the covariance or relationship among the features in \(X\) is proportional to the \(p \times p\) matrix

\[X^TX\]

Further, we can express this covariance structure as

\[X^TX = W \Lambda W^T\]

where \(W\) is a \(p \times p\) matrix, the columns of which are eigenvectors of \(X^TX\), and \(\Lambda\) is a diagonal matrix of eigenvalues.

  • principal components
    The \(p\) principal components are defined by the columns of \(W\).
  • scores
    \(XW\) maps the original data to the transformed data set, the columns of which are the scores (coordinates) for each of the PCs.
  • loadings
    Scaling the columns of \(W\), ie. multiplying them by the square root of the corresponding eigenvalue, gives the loadings.
  • proportion of variance explained by each principal component
    The eigenvalues in \(\Lambda\) measure the relative proportion of variability that’s explained by each principal component (eg: \(\lambda_1 / (\lambda_1 + \cdots + \lambda_p)\)).

Alternatively, we can obtain the principal components via singular value decomposition (SVD). Instead of decomposing \(X^TX\), SVD decomposes \(X\). In fact, this is a bit more computationally stable! Specifically, SVD expresses \(X\) as

\[X = U \Sigma W^T\]

where \(U\) is an \(n \times n\) matrix of orthogonal left singular vectors of \(X\) and \(\Sigma\) is an \(n \times p\) diagonal matrix of singular values. Then \(W\) still provides the principal components and the diagonal of \(\Sigma^2\) is equivalent to the diagonal of \(\Lambda\) since by the SVD,

\[X^TX = W \Sigma^T U^T U \Sigma W^T = W \Sigma^2 W^T\]

Further resource: https://www.hackerearth.com/blog/developers/principal-component-analysis-with-linear-algebra/